Settings (1-4) do not contain the two-step disentanglement architecture and are similar to our 2nd step of training but without the reconstruction loss. Setting (1) uses a standard baseline classifier with only \(\mathcal {L}_y\)  as a loss. Setting (2) is a classifier trained with \(\mathcal {L}_y\)  and an adversarial loss \(\mathcal {L}_{adv}\) . This is similar to [1]}.
In (3) and (4) we consider these variants with our local fairness loss (\(\mathcal {L}_{local}\) ).
As can be seen, classifier (1) is very biased, archiving \(67\%\)  accuracy on the sensitive variable, and only \(57\%\)  accuracy on the true labels \(y\)  of the test set (recall that the train and test distributions differ). Adding an adversarial loss (\(\mathcal {L}_{adv}\) ) in (2) improves the bias of the representation by reducing the accuracy to sensitive variable to \(55\%\) , while also improving the prediction accuracy to the true test labels \(y\) . Adding local fairness (\(\mathcal {L}_{local}\) ) in (3) and (4) reduces both results towards random chance. This indicates that the classifier is unable to learn and gives a random output, showing that in the standard setting, without disentanglement, local fairness does not improve fairness.
